https://r2.erweima.ai/mathbot/image/17255118653949255ip9q.jpg

The image shows the general formula for an absolute value function, given by:

\[
f(x) = a|x - h| + k
\]

This is the standard form of an absolute value function, where:
- \( a \) controls the vertical stretch or compression (and reflection if \( a \) is negative),
- \( h \) shifts the graph horizontally (to the right if positive, to the left if negative),
- \( k \) shifts the graph vertically (up if positive, down if negative).

There is also an example function given:

\[
f(x) = 2|x - 3| - 1
\]

This specific example has:
- \( a = 2 \), indicating the graph is stretched vertically by a factor of 2,
- \( h = 3 \), shifting the graph 3 units to the right,
- \( k = -1 \), shifting the graph 1 unit down.

Would you like further details or explanations?

### 5 related questions:
1. How does the value of \( a \) affect the graph of the absolute value function?
2. What happens if \( a \) is negative in the absolute value function?
3. How would you graph the function \( f(x) = |x - 2| + 4 \)?
4. Can you derive the vertex of an absolute value function based on \( h \) and \( k \)?
5. How does changing \( h \) impact the graph's horizontal position?

**Tip:** Always check the sign of \( a \) and \( h \) carefully when graphing transformations of functions!

What is the rule of correspondence for the function f(x) = a|x - h| + k and how does it transform the graph?

Explore the standard form of the absolute value function f(x) = a|x - h| + k, including how the parameters a, h, and k transform the graph. Learn about vertical stretches, compressions, and shifts for students in Grades 8-10.

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